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We have a $48$-inch-by-$80$-inch wooden door with two rectangular glass windows, not necessarily of the same size. The wooden areas of the door are of uniform width, labeled "$x$" in the picture below. The glass occupies $1590$ square inches. Find the width of the wooden areas.

I'm assuming that you would do $(48-2x)(80-3x)$ which would simplify to $3x^{2}-152x+1920$.

I'm really close on the factoring. I don't know what quadratic formula is yet or how to really use it.

The area of the wood can be broken into the two rectangles up the sides, which are $80 times x$ and three rectangles in the middle, which are $(48-2x) times x$. The total are of wood is then $2cdot 80 cdot x + 3 cdot (48-2x) x=-6x^2+304x$ This plus the glass area equals the total door area, so
$$-6x^2+304x+1590=48cdot 80=3840\0=6x^2-304x+2250$$

Initially, you gave a correct expression for the area of the glass: $$(48-2x)(80-3x) tag{1}$$
"Simplifying" to $3x^2-152x+1920$, however, is incorrect. You seem to have taken the expanded form of $(1)$ ...

$$6 x^2 - 304 x + 3840 tag{2}$$

... and divided-through by $2$. But $(2)$ is an expression, representing some number. You cannot "simplify" an expression by dividing it by something. (If I have $30$ apples, you can't "simplify" my inventory by dividing by $2$ to say that I only have $15$ apples.)

The next step should be to take your calculated glass area, and equate it with the given glass area:
$$6x^2-304x+3840 = 1590 tag{3}$$
Here, you are allowed to divide-through by $2$ if you like (it helps to make the numbers smaller), provided that you do so to both sides.
$$3x^2-152x+1920 = 795 tag{4}$$
(If I have $30$ apples, then it's perfectly valid to say that half my inventory is half of $30$; that is, half my inventory is $15$. (Note how this situation is different from the one I described after $(2)$.) So, here, it's valid to say that half the calculated area of glass is half the given area of glass.)

From here, we can re-write (subtracting $795$ from both sides) ...
$$3x^2-152x+1125 = 0 tag{5}$$
... and then hope to factor the left-hand side. Can you proceed from here?